On the Linearization of Second-Order Differential and Difference Equations

Дородницын, В. А.

doi:10.3842/sigma.2006.065

Cited by 3 publications

(3 citation statements)

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“…The open problem is to find covariant conditions for the linearization of equations whose Lie symmetry algebra is not the sl(3, R). As it has been shown [27,28], non-point transformations achieve the linearization of certain equations of this type. Therefore, it is reasonable to assume that a proper covariant condition could be in the jet bundle of the affine space spanned by the variables of the equation.…”

Section: Discussionmentioning

confidence: 95%

Linearization of Second-Order Non-Linear Ordinary Differential Equations: A Geometric Approach

Tsamparlis

2023

Symmetry

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Using the coefficients of a system semilinear cubic in the first derivative second order differential equations one defines a connection in the space of the independent and dependent variables, which is specified modulo two free parameters. In this way, to any such equation one associates an affine space which is not necessarily Riemannian, that is, a metric is not required. If such a metric exists, then under the Cartan parametrization the geodesic equations of the metric coincide with the system of the considered semilinear equations. In the present work, we consider semilinear cubic in the first derivative second order differential equations whose Lie symmetry algebra is the sl(3,R). The covariant condition for these equations is the vanishing of the curvature tensor. We demonstrate the method in the solution of the Painlevé-Ince equation and in a system of two equations. Because the approach is geometric, the number of equations in the system is not important besides the complication in the calculations. It is shown that it is possible to linearize an equation in this form using a different covariant condition, for example, assuming the space to be of constant non-vanishing curvature. Finally, it is shown that one computes the associated metric to a semilinear cubic in the first derivatives differential equation using the inverse transformation derived from the transformation of the connection.

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Section: Discussionmentioning

confidence: 95%

Linearization of Second-Order Non-Linear Ordinary Differential Equations: A Geometric Approach

Tsamparlis

2023

Symmetry

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“…ãäå ôóíêöèÿ ψ = ψ(t) óäîâëåòâîðÿåò óðàâíåíèþ äâèaeåíèÿ ÷àñòèöû â ïîòåíöèàëüíîì ïîëå [37] ψ ψ 2 + 12 = 0.…”

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“…Ïîñëåäíåå óðàâíåíèå ìîaeåò áûòü ëèíåàðèçîâàíî ñ ïîìîùüþ êàñàòåëüíîãî ïðåîáðàçîâàíèÿ (ñì. ïîäðîáíîñòè â[37]), íî åãî îáùåå ðåøåíèå ÷åðåç ýëåìåíòàðíûå ôóíêöèè íå âûðàaeàåòñÿ. Òàêaeå äëÿ óðàâíåíèÿ(35) èçâåñòåí ïåðâûé…”

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